Abstract
This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.
Highlights
We are interested in the long time behavior of isolated biological populations with a regulated reproduction
We will study the long time behavior of the process conditioned on non extinction and the related notion of quasi-stationarity
It is clear that any Yaglom limit and any quasi-stationary distribution (QSD) is a quasi-limiting distribution (QLD)
Summary
We are interested in the long time behavior of isolated biological populations with a regulated (density-dependent) reproduction. The case of Markov processes on finite state spaces has been studied by Darroch and Seneta, who proved under some irreducibility conditions the existence and uniqueness of the QSD, for both discrete [18] and continuous time settings [19] (detailed proofs and results are reproduced in Section 3 of the present paper). An important question is to know whether the convergence to the Yaglom limit happens before the typical time of extinction, or if it happens only after very large time periods, in which case the populations whose size are distributed with respect to the Yaglom limit are very rare Both situations can appear, as illustrated by the simple example of Section 2.3.
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